solve te initial value problems

Question

dr/dtheta= 3cos^2 (pi/4-theta), r(0)=pi/8

Answer 1

The main point here is to use some trig identities on the right-hand side to produce a function that we know how to integrate. Here is a sketch of the solution. Please let me know if you have any questions.

First, use the addition law for cosine to get

cos(pi/4 - theta) = cos(theta)/sqrt(2) + sin(theta)/sqrt(2)

Now square the result (and use cos^2(theta) + sin^2(theta) = 1 and the double-angle formula for sine) to get

cos^2(pi/4 - theta) = 1/2 + cos(theta) sin(theta)

= 1/2 + (1/2) sin(2 theta)

Now integrate 3 times the right-hand side and substitute r(0) = pi/8 to find the constant of integration.

Don't forget to check your answer by differentiating.